Study Guide for Midterm
1. Using the circle shown,
a) Find the center of the circle:
4
(1, 5/2)
2
b) Find the radius of the circle:
-5
5
(-3, -1/2)
-2
c) Write the equation for the circle:
-4
2. The equation for a circle is given by: 𝑥 2 + 𝑦 2 − 4𝑥 + 6𝑦 + 1 = 0. By writing this equation in standard
form, find the:
a) Center of the circle
b) Radius of the circle
3. Answer the following.
(a) Sketch and label the angle   
5
in the coordinate plane.
6

(b) Find one positive and one negative angle coterminal with the angle   580 .
(c) Convert 5 radians to degrees.
(d) Convert 100° to radians.
(e) Give a point on the unit circle corresponding to   

2
.
(f) What is the sign of csc( ) if the terminal side of  is in Quadrant III?
(g) What is the period of f (t )  cos( t ) ? What is the domain of f (t )  cos( t ) ?
(h) What is the period of f (t )  sec( t ) ? What is the domain of f (t )  sec( t ) ?
4. The central angle   210 forms an arc of length 9 cm. What is the radius of the circle? Draw a picture (neatly)
representing this problem, labeling the radius, central angle, and arc length.
Study Guide for Midterm
5. Solve the right triangle and find the specified trigonometric functions. Give exact values.
 = _______
 = _______
c = _______
sin  = _______
cos  = _______
tan  = _______
csc = _______
sec = _______
cot  = _______
𝛽
𝑐
7
𝛼
7√3
6. Solve the right triangle and find the specified trigonometric functions. Give exact values.
 = _______
c = _______
33.7
c
sin  = _______
3
cos  = _______
tan  = _______

2
1
4

7. Find the value of the six trigonometric functions given that P   ,
with  in standard position. Sketch  in standard position.
5
 is on the terminal side of the angle  ,
2 
8. Given the point is on the unit circle, complete the ordered pair x, y  for the quadrant indicated.
a.
1 
 , y  ; QIV
3 
b.

95 


 12 , y  ; QII


9. Find two angles in the interval 0, 2  that satisfy sin   
3
.
2
10. Given that tan( )  6 and csc( )  0 , then  is in Quadrant ______. Determine the value of the other five
trigonometric functions.
11. State the quadrant of the terminal side of  . Then state the value of the six trig functions of  .
a.
cos  
4
; sin   0
5
b.
tan   
12
; cos  0
5
Study Guide for Midterm
12. Evaluate the following. Give exact answers.
 
sin   = _______
6


cos 600 = _______
 
csc    = _______
 4
tan 480 = _______
 
sec   = _______
2
 
tan    = _______
 3
 11
sec  
 6
csc   = _______



 = _______

13. At a certain time of day, the Washington Monument casts a shadow 700 feet long. From the tip of the shadow, the
angle from the horizontal to the top of the monument is 30 . Use this information to find the height of the
monument.
14. From the observation deck of a seaside building 120 m high, Armando sees two fishing boats in the distance. The
angle of depression to the nearer boat is 45 , and it is 30 for the other boat. How far out to sea is the nearer boat?
How far apart are the two boats? (See figure below.)
Armando
30
45
15. From a hotel room window, a tourist sees some window washers high above on the building across the street. The
tourist estimates that the buildings are 50 ft. apart, the angle of elevation to the workers is about 45 , and that the
angle of depression to the base of the building across the street is 60 . How high above the ground is the window
of the tourist’s hotel room? How high above ground are the workers?


16. For the function f t   5 sin  3t 

  2 determine the following information.
2
Amplitude: _____________
Domain: ___________________
Period: ________________
Range: ____________________
Horizontal Shift: _________
Primary interval: ____________
Vertical Shift: ___________
y-intercept: ________________
17. State the domain, range, and period of the given function, then sketch two complete periods.
a.
1 
g t   3 cot  t 
2 
b.
1 
ht   2 cos t 
3 
c.
k t  
3
csct 
2
Study Guide for Midterm
18. Determine the equation for the graph given, and state the domain of the function.
f t   _______________________________
Domain: _______________________________